Question

An approximate partition function \(Q(N, V, T)\) of a gas is given below. \[ Q(N, V, T) = \frac{1}{N!} \left( \frac{2 \pi m k_B T}{h^2} \right)^{3N/2} (V - Nb)^N \] The equation of state(s) for this gas is/are [Note: \(b\) is a parameter independent of volume.]

• A. \(P(V - Nb) = Nk_BT\)
• B. \(PV(N - b) = k_BT\)
• C. \(PV = Nk_BT\)
• D. \(P(V - Nb) = Nk_B\)

Hint:

The equation of state of a gas can be derived from its canonical partition function by using the relation \(P = k_B T (\frac{\partial \ln Q}{\partial V})_{N, T}\). Remember to correctly apply the rules of differentiation and logarithms during the derivation. The term \((V - Nb)\) in the partition function accounts for the excluded volume of the gas molecules.

Answer 1

The Correct Option is A

The equation of state can be derived from the partition function using the relation: \[ P = k_B T \left( \frac{\partial \ln Q(N, V, T)}{\partial V} \right)_{N, T} \] First, let's take the natural logarithm of the partition function \(Q(N, V, T)\): \[ \ln Q(N, V, T) = \ln \left[ \frac{1}{N!} \left( \frac{2 \pi m k_B T}{h^2} \right)^{3N/2} (V - Nb)^N \right] \] Using the properties of logarithms, we can expand this: \[ \ln Q(N, V, T) = -\ln(N!) + \frac{3N}{2} \ln \left( \frac{2 \pi m k_B T}{h^2} \right) + N \ln(V - Nb) \] Now, we need to take the partial derivative of \( \ln Q \) with respect to \( V \) at constant \( N \) and \( T \): \[ \left( \frac{\partial \ln Q(N, V, T)}{\partial V} \right)_{N, T} = \frac{\partial}{\partial V} \left[ -\ln(N!) + \frac{3N}{2} \ln \left( \frac{2 \pi m k_B T}{h^2} \right) + N \ln(V - Nb) \right]_{N, T} \] The first two terms do not depend on \( V \), so their derivatives with respect to \( V \) are zero. The derivative of the last term is: \[ \frac{\partial}{\partial V} [N \ln(V - Nb)] = N \frac{1}{V - Nb} \frac{\partial}{\partial V} (V - Nb) = N \frac{1}{V - Nb} (1 - 0) = \frac{N}{V - Nb} \] Now, substitute this into the equation for pressure: \[ P = k_B T \left( \frac{N}{V - Nb} \right) \] Rearranging this equation gives the equation of state for the gas: \[ P(V - Nb) = Nk_B T \] This equation is similar to the van der Waals equation, where \(Nb\) represents the excluded volume due to the finite size of the gas molecules. Comparing this derived equation of state with the given options, we find that it matches option (A).